Irreducible Polynomials

**Coda202** · April 27th 2009, 05:01 PM

let a= sqrt(7 + 4 *sqrt(3)) find irr(a,Q)

now let a= sqrt(7) + i find irr(a,Q)

**NonCommAlg** · April 28th 2009, 03:48 AM

Originally Posted by Coda202

let a= sqrt(7 + 4 *sqrt(3)) find irr(a,Q)

$\sqrt{7 + 4\sqrt{3}}=2 + \sqrt{3}.$

now let a= sqrt(7) + i find irr(a,Q)

$a^4 - 12a^2 + 64=0.$ show that the polynomial $x^4-12x^2+64$ is irreducible over $\mathbb{Z},$ and therefore over $\mathbb{Q}.$ in order to show this you only need to prove that we cannot have

$x^4-12x^2+64=(x^2+ax+b)(x^2+cx+d),$ for some integers $a,b,c,d.$ that is very easy to verify!

**mathman88** · April 28th 2009, 01:51 PM

Originally Posted by NonCommAlg

Show that the polynomial $x^4-12x^2+64$ is irreducible over $\mathbb{Z},$ and therefore over $\mathbb{Q}.$

How does irreducibile in $\mathbb{Z}$ imply irreducible in $\mathbb{Q}$ ?

**NonCommAlg** · April 28th 2009, 02:16 PM

Originally Posted by mathman88

How does irreducibile in $\mathbb{Z}$ imply irreducible in $\mathbb{Q}$ ?

Gauss's lemma.

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